Free math worksheets

- Which angle pairs are formed by a transversal crossing parallel lines? - The two marked angles lie inside the parallel lines on opposite sides of the transversal. - What is true about alternate interior angles?

1. The given \(62^\circ\) angle and \(\beta\) are alternate interior angles. 2. Alternate interior angles are congruent when the lines are parallel. 3. Therefore, \(\beta = 62^\circ\).

\(\beta = 62^\circ\)

- What angle relationship is formed when a transversal crosses two parallel lines? - Identify the alternate interior angle corresponding to \(\beta\).

1. Angles \(\beta\) and \(\varepsilon\) are alternate interior angles formed by a transversal crossing parallel lines. 2. Therefore, \(\varepsilon = \beta = 68^\circ\).

\(\varepsilon = 68^\circ\)

- What relationship do consecutive angles in a rhombus have? - Use that relationship with the given \(74^\circ\) angle.

1. Consecutive angles in a rhombus are supplementary, so \(\alpha + \beta = 180^\circ\). 2. Substitute \(\alpha = 74^\circ\): \(74^\circ + \beta = 180^\circ\). 3. \(\beta = 106^\circ\).

\(\beta = 106^\circ\)

- What must the interior angles of a triangle total? - Add the three given measures and compare the result with that total.

1. The interior angles of any triangle must sum to \(180^\circ\). 2. The given measures sum to \(110^\circ + 45^\circ + 35^\circ = 190^\circ\). 3. Since \(190^\circ \ne 180^\circ\), no triangle can have all three given angle measures.

No. The given angles total \(190^\circ\), not \(180^\circ\).

- What is the sum of same-side interior angles when the lines are parallel? - Use the given \(110^\circ\) angle with that relationship.

1. Same-side interior angles formed by a transversal of parallel lines are supplementary. 2. The other angle measures \(180^\circ - 110^\circ = 70^\circ\).

The other angle measures \(70^\circ\).

- What is the relationship between consecutive angles in a parallelogram? - Apply that relationship to \(\angle A\) and \(\angle B\).

1. Consecutive angles in a parallelogram are supplementary. 2. Therefore, \(\beta = 180^\circ - 55^\circ = 125^\circ\).

\(\beta = 125^\circ\)

- What total do complementary angles have? - Subtract the known angle from that total.

1. Complementary angles have a sum of \(90^\circ\). 2. The missing angle is \(90^\circ-28^\circ=62^\circ\).

\(62^\circ\)

- What total is formed by a linear pair? - Which vocabulary word describes two angles whose measures have that total?

1. A linear pair has a sum of \(180^\circ\). 2. \(\beta=180^\circ-117^\circ=63^\circ\). 3. Because their measures sum to \(180^\circ\), the angles are supplementary.

\(\beta=63^\circ\). The angles are supplementary.

- What relationship do corresponding angles have when the lines are parallel? - What is the sum of a linear pair? - Use vertical and corresponding angle relationships to count all congruent angles.

1. Corresponding angles formed by a transversal of parallel lines are congruent. Therefore, each angle measures \(156^\circ \div 2 = 78^\circ\). 2. Angles in a linear pair are supplementary, so the adjacent angle measures \(180^\circ - 78^\circ = 102^\circ\). 3. Vertical, corresponding, and alternate angles create a set of four congruent acute angles and four congruent obtuse angles. Therefore, four of the eight angles measure \(78^\circ\).

a) \(78^\circ\) b) \(102^\circ\) c) Four angles

- Recall the angle-pair names used with parallel lines and a transversal. - What is the sum of a linear pair? - What happens to a linear pair when one angle is \(90^\circ\)?

1. Three other \(65^\circ\) angles are the vertical angle to \(\alpha\), the corresponding angle at the other intersection, and the vertical angle to that corresponding angle. The last of these is also alternate to \(\alpha\). 2. Since a linear pair totals \(180^\circ\), \(\beta = 180^\circ - 65^\circ = 115^\circ\). 3. If the transversal is perpendicular to the parallel lines, one angle measures \(90^\circ\). Its linear-pair angle also measures \(180^\circ - 90^\circ = 90^\circ\), and vertical and corresponding angles are congruent. Therefore, all eight angles measure \(90^\circ\), so the claim is true.

a) The vertical angle to \(\alpha\), its corresponding angle, and the vertical angle to that corresponding angle all measure \(65^\circ\). b) \(\beta = 115^\circ\) c) True. All eight angles would measure \(90^\circ\).

- Plot the points before deciding the direction of each ray. - Compare the horizontal and vertical coordinate changes from \(S\). - Identify horizontal, vertical, and equal-change diagonal directions.

1. Ray \(\overrightarrow{SA}\) is horizontal and points to the right. 2. Ray \(\overrightarrow{SB}\) is vertical and points upward, so it is perpendicular to \(\overrightarrow{SA}\). The angle is \(90^\circ\). 3. From \(S\) to \(C\), the horizontal and vertical changes are both \(4\), so \(\overrightarrow{SC}\) follows a \(45^\circ\) diagonal. The smaller angle is \(45^\circ\). 4. Ray \(\overrightarrow{SD}\) is vertical and points downward, so it is perpendicular to \(\overrightarrow{SA}\). The smaller angle is \(90^\circ\).

a) \(90^\circ\) b) \(45^\circ\) c) \(90^\circ\)

- First classify each pictured angle relative to \(90^\circ\) and \(180^\circ\). - A complement completes \(90^\circ\), while a supplement completes \(180^\circ\). - For the reflex angle, compare its measure with a straight angle rather than trying to form a positive supplement.

1. From the diagrams, \(\alpha\) is acute, \(\beta\) is obtuse, and \(\gamma\) is reflex, so the matching measures are \(40^\circ\), \(135^\circ\), and \(220^\circ\). 2. The complement of \(\alpha\) is \(90^\circ-40^\circ=50^\circ\). 3. The supplement of \(\beta\) is \(180^\circ-135^\circ=45^\circ\), which is acute. 4. Angle \(\gamma\) exceeds a straight angle by \(220^\circ-180^\circ=40^\circ\).

\(\alpha=40^\circ\), \(\beta=135^\circ\), \(\gamma=220^\circ\) a) \(50^\circ\) b) \(45^\circ\) c) \(40^\circ\)

- What symmetry property does an isosceles trapezoid have? - How are consecutive interior angles related when lines are parallel? - Work along one leg at a time.

1. Base angles of an isosceles trapezoid are congruent, so \(\beta = \alpha = 65^\circ\). 2. Consecutive interior angles along each leg are supplementary because \(AB \parallel CD\). 3. Therefore, \(\delta = 180^\circ - 65^\circ = 115^\circ\). 4. Likewise, \(\gamma = 180^\circ - 65^\circ = 115^\circ\).

\(\beta = 65^\circ\), \(\gamma = 115^\circ\), and \(\delta = 115^\circ\).

- What is true about opposite angles in a parallelogram? - How are consecutive angles in a parallelogram related? - You may also use the quadrilateral angle sum.

1. Opposite angles of a parallelogram are congruent, so \(\delta = \beta = 120^\circ\). 2. Consecutive angles of a parallelogram are supplementary. 3. Therefore, \(\alpha = 180^\circ - 120^\circ = 60^\circ\). 4. Since opposite angles are congruent, \(\gamma = \alpha = 60^\circ\).

\(\alpha = 60^\circ\), \(\gamma = 60^\circ\), and \(\delta = 120^\circ\).

- Which sides of the trapezoid are parallel? - How are same-side interior angles related when lines are parallel? - Work along each leg separately.

1. Consecutive interior angles along each leg of the trapezoid are supplementary because \(AB \parallel CD\). 2. Along leg \(AD\), \(\delta = 180^\circ - 70^\circ = 110^\circ\). 3. Along leg \(BC\), \(\beta = 180^\circ - 135^\circ = 45^\circ\).

\(\beta = 45^\circ\) and \(\delta = 110^\circ\).

- What angle relationships are formed when a transversal cuts parallel lines? - Identify the angle corresponding to the given angle. - What is the sum of a linear pair?

1. The angle corresponding to the given \(55^\circ\) angle also measures \(55^\circ\). 2. That corresponding angle and \(\alpha\) form a linear pair. 3. Therefore, \(\alpha = 180^\circ - 55^\circ = 125^\circ\).

\(\alpha = 125^\circ\)

- Classify each angle by comparing it with a right angle and a straight angle. - A positive complement is possible only for an angle smaller than \(90^\circ\). - A positive supplement is possible only for an angle smaller than \(180^\circ\).

1. Angle \(\alpha\) is acute, \(\beta\) is obtuse, and \(\gamma\) is reflex. 2. Only an angle less than \(90^\circ\) can have a positive complement, so only \(\alpha\) can. 3. Any angle less than \(180^\circ\) can have a positive supplement, so \(\alpha\) and \(\beta\) can. 4. Since \(\gamma>180^\circ\), subtracting it from either \(90^\circ\) or \(180^\circ\) would give a negative value, so it has neither a positive complement nor a positive supplement.

\(\alpha\): acute; \(\beta\): obtuse; \(\gamma\): reflex a) \(\alpha\) only b) \(\alpha\) and \(\beta\) c) \(\gamma>180^\circ\), so neither \(90^\circ-\gamma\) nor \(180^\circ-\gamma\) is positive.

- How are consecutive interior angles in a parallelogram related? - What total must the angles at \(A\) and \(D\) have? - Write an equation containing \(\alpha\).

1. Consecutive interior angles of a parallelogram are supplementary because opposite sides are parallel. 2. The angles at \(A\) and \(D\) are consecutive, so \(72^\circ + \alpha = 180^\circ\). 3. Therefore, \(\alpha = 180^\circ - 72^\circ = 108^\circ\).

\(\alpha = 108^\circ\)

- A vertical and horizontal grid line form \(90^\circ\). - A square’s diagonal divides a right angle into two equal angles. - Pay attention to the direction in which the reflex angle is marked.

1. In a), the diagonal of a square divides a right angle in half, so \(\alpha=45^\circ\). 2. In b), the ray points along the northwest diagonal, so \(\beta=90^\circ+45^\circ=135^\circ\). 3. In c), the angle is measured counterclockwise from the right-pointing ray to the downward ray, which is three-fourths of a full turn: \(\gamma=270^\circ\). 4. Therefore, \(\alpha\) is acute, \(\beta\) is obtuse, and \(\gamma\) is reflex.

a) \(\alpha=45^\circ\), acute b) \(\beta=135^\circ\), obtuse c) \(\gamma=270^\circ\), reflex

- Which pair of sides is parallel? - What angle relationships are created when a transversal crosses parallel lines? - Look for corresponding angles and same-side interior angles. - Compare each unknown angle with a given angle along the same transversal.

1. Because \(AB \parallel CD\), \(\alpha\) and the \(70^\circ\) angle are corresponding angles formed by transversal \(AD\). Therefore, \(\alpha = 70^\circ\). 2. The \(60^\circ\) angle and \(\beta\) are same-side interior angles formed by transversal \(BC\), so they are supplementary. Therefore, \(\beta = 180^\circ - 60^\circ = 120^\circ\).

\(\alpha = 70^\circ\) and \(\beta = 120^\circ\)

- Recall the angle relationships formed when a transversal crosses parallel lines. - How are \(\angle A\) and \(\delta\) related when \(AB \parallel CD\)? - What is the sum of the two same-side interior angles along either leg?

1. Because \(AB \parallel CD\), the same-side interior angles along each leg of the trapezoid are supplementary. 2. Along leg \(AD\), \(\delta = 180^\circ - 75^\circ = 105^\circ\). 3. Along leg \(BC\), \(\gamma = 180^\circ - 55^\circ = 125^\circ\).

\(\gamma = 125^\circ\) and \(\delta = 105^\circ\)

- Identify the corresponding angles formed by the transversal. - Which angles are vertical angles? - Which adjacent angles form a linear pair? - Use the fact that the transversal is not perpendicular to the parallel lines.

1. Angles \(\alpha\) and \(\beta\) are corresponding angles, so \(\alpha = \beta\). 2. Angles \(\alpha\) and \(\gamma\) are vertical angles, so \(\alpha = \gamma\). 3. Angles \(\beta\) and \(\delta\) form a linear pair, so \(\beta + \delta = 180^\circ\). 4. Since \(\gamma = \alpha = \beta\), the equation \(\gamma = \delta\) could hold only if both angles measured \(90^\circ\). The transversal is not perpendicular to the parallel lines, so statement d) is false.

d) \(\gamma = \delta\) is false.

- What is the sum of the interior angles of a triangle? - Which two angle measures are already given in the diagram?

1. The two base angles of the triangle measure \(50^\circ\) and \(75^\circ\). 2. By the triangle angle sum, \(\gamma = 180^\circ - 50^\circ - 75^\circ = 55^\circ\).

\(\gamma = 55^\circ\)

- What angle measure forms a straight line? - How are adjacent angles at an intersection related? - What is true about opposite angles at an intersection?

1. Angle \(\beta\) forms a linear pair with the \(142^\circ\) angle, so \(\beta = 180^\circ - 142^\circ = 38^\circ\). 2. Angle \(\gamma\) is vertical to the \(142^\circ\) angle, so \(\gamma = 142^\circ\). 3. Angle \(\delta\) is vertical to \(\beta\), so \(\delta = 38^\circ\). 4. Opposite angles formed by two intersecting lines are called vertical angles.

\(\beta = 38^\circ\), \(\gamma = 142^\circ\), and \(\delta = 38^\circ\). The opposite angle pairs are vertical angles.

- What angle relationships occur when a transversal crosses parallel lines? - Which angles must have equal measures? - What are adjacent angles that form a straight line called? - Review corresponding, alternate interior, vertical, and supplementary angles.

1. Angle \(\alpha\) is alternate interior to the given \(115^\circ\) angle, so \(\alpha = 115^\circ\). 2. Angles \(\alpha\) and \(\beta\) form a linear pair, so \(\beta = 180^\circ - 115^\circ = 65^\circ\). 3. Angle \(\gamma\) forms a linear pair with the given \(115^\circ\) angle, so \(\gamma = 65^\circ\).

\(\alpha = 115^\circ\), \(\beta = 65^\circ\), and \(\gamma = 65^\circ\)

- What angle pairs are formed when a transversal crosses two parallel lines? - Review corresponding angles and linear pairs. - What is the sum of two angles that form a straight line?

1. Angle \(\beta\) corresponds to \(\alpha\). Corresponding angles formed by a transversal crossing parallel lines are congruent, so \(\beta = 65^\circ\). 2. Angles \(\beta\) and \(\gamma\) form a linear pair. Therefore, \(\gamma = 180^\circ - 65^\circ = 115^\circ\).

\(\beta = 65^\circ\) and \(\gamma = 115^\circ\)

- Review vertical angles and linear pairs at an intersection. - Which angle relationships occur when a transversal crosses parallel lines? - What is the sum of a linear pair?

1. Angle \(\beta\) is vertical to the given \(54^\circ\) angle, so \(\beta = 54^\circ\). 2. Angle \(\gamma\) corresponds to the given \(54^\circ\) angle, so \(\gamma = 54^\circ\). 3. Angles \(\gamma\) and \(\delta\) form a linear pair. Therefore, \(\delta = 180^\circ - 54^\circ = 126^\circ\).

\(\beta = 54^\circ\), \(\gamma = 54^\circ\), and \(\delta = 126^\circ\)

- What angle pairs are formed when a transversal crosses parallel lines? - What is the sum of adjacent angles that form a straight angle? - Which unknown angles are related directly to the given angles?

1. The angles \(50^\circ\), \(\alpha\), and \(60^\circ\) form a straight angle along line \(g\). Therefore, \(\alpha = 180^\circ - 50^\circ - 60^\circ = 70^\circ\). 2. Angle \(\beta\) is alternate interior to the given \(50^\circ\) angle, so \(\beta = 50^\circ\). 3. Angle \(\gamma\) is supplementary to the angle corresponding to the given \(60^\circ\) angle. Therefore, \(\gamma = 180^\circ - 60^\circ = 120^\circ\).

\(\alpha = 70^\circ\), \(\beta = 50^\circ\), and \(\gamma = 120^\circ\)

- What angle pairs are formed by two intersecting lines? - Which angles must be congruent when a transversal crosses parallel lines? - Are the marked angles opposite at one intersection or in matching positions at two intersections?

1. Angle \(\beta\) is vertical to \(\alpha\), so \(\beta = 68^\circ\). 2. Angle \(\gamma\) corresponds to \(\alpha\), so \(\gamma = 68^\circ\).

\(\beta = 68^\circ\) because it is vertical to \(\alpha\). \(\gamma = 68^\circ\) because it corresponds to \(\alpha\).

- Look for corresponding or alternate interior angles formed by the parallel lines. - What is the sum of adjacent angles that form a straight line? - Which angles must be congruent because \(g \parallel h\)?

1. Angle \(\alpha\) is alternate interior to the given \(36^\circ\) angle, so \(\alpha = 36^\circ\). 2. Angle \(\gamma\) is alternate interior to the given \(68^\circ\) angle, so \(\gamma = 68^\circ\). 3. The angles \(68^\circ\), \(\beta\), and \(36^\circ\) form a straight angle on line \(g\). Therefore, \(\beta = 180^\circ - 68^\circ - 36^\circ = 76^\circ\).

\(\alpha = 36^\circ\), \(\beta = 76^\circ\), and \(\gamma = 68^\circ\)

- What is the sum of two angles that form a linear pair? - Treat the two vertices independently; the quadrilateral angle sum is not needed.

1. Since \(\alpha\) and \(100^\circ\) form a linear pair, \(\alpha = 180^\circ - 100^\circ = 80^\circ\). 2. Since \(\delta\) and \(85^\circ\) form a linear pair, \(\delta = 180^\circ - 85^\circ = 95^\circ\).

\(\alpha = 80^\circ\) and \(\delta = 95^\circ\)

- What is the total measure of a linear pair? - Each unknown angle is adjacent to a \(42^\circ\) angle on a straight line.

1. Each unknown angle and an adjacent \(42^\circ\) angle form a linear pair, so their measures sum to \(180^\circ\). 2. \(\alpha = 180^\circ - 42^\circ = 138^\circ\). 3. Likewise, \(\beta = 180^\circ - 42^\circ = 138^\circ\).

\(\alpha = 138^\circ\) and \(\beta = 138^\circ\)

- Locate the inward-pointing vertex to identify the reflex interior angle. - Decide which marked angles are below \(90^\circ\) and which are below \(180^\circ\). - Connect those comparisons to the definitions of complement and supplement.

1. The inward vertex is \(Q\), so \(\phi_2\) is greater than \(180^\circ\) and is reflex. 2. Angles \(\phi_1\), \(\phi_3\), and \(\phi_4\) are acute, so each is less than \(90^\circ\). 3. Therefore, \(\phi_1\), \(\phi_3\), and \(\phi_4\) each have both a positive complement and a positive supplement. 4. Since \(\phi_2>180^\circ\), it has neither a positive complement nor a positive supplement.

a) \(\phi_2\) b) \(\phi_1,\phi_3,\phi_4\) c) \(\phi_1,\phi_3,\phi_4\) d) \(\phi_2>180^\circ\), so subtracting it from \(90^\circ\) or \(180^\circ\) cannot give a positive angle measure.

- Which angles are vertical when two lines intersect? - What is the sum of the measures in a linear pair?

1. Vertical angles are congruent, so the opposite angle also measures \(32^\circ\). 2. Each adjacent angle forms a linear pair with the given angle, so its measure is \(180^\circ - 32^\circ = 148^\circ\). 3. Therefore, both adjacent angles measure \(148^\circ\).

The vertical angle measures \(32^\circ\), and each adjacent angle measures \(148^\circ\).

- What is the total angle measure around a point? - First find one ninth of that total. - Then use the relationship between angles in a linear pair.

1. The four angles around the intersection total \(360^\circ\). 2. Find one ninth of the total: \(\alpha = \frac{1}{9} \cdot 360^\circ = 40^\circ\). 3. An adjacent angle forms a linear pair with \(\alpha\), so its measure is \(180^\circ - 40^\circ = 140^\circ\).

\(\alpha = 40^\circ\), and an adjacent angle measures \(140^\circ\).

- What relationship do corresponding angles have when the lines are parallel? - What is the sum of the measures in a linear pair?

1. The corresponding obtuse angle at line \(h\) also measures \(115^\circ\). 2. That angle and \(\beta\) form a linear pair, so \(\beta = 180^\circ - 115^\circ = 65^\circ\).

\(\beta = 65^\circ\)

- Identify the two pairs of parallel sides in the parallelogram. - Which alternate interior angles are formed by diagonal \(QS\)?

1. Opposite sides of a parallelogram are parallel: \(PQ \parallel RS\) and \(PS \parallel QR\). 2. Diagonal \(QS\) is a transversal. Alternate interior angles give \(\gamma = \alpha = 35^\circ\) and \(\delta = \beta = 45^\circ\).

\(\gamma = 35^\circ\) and \(\delta = 45^\circ\)

- Translate “complementary” into an equation about the sum of the two measures. - Solve for \(x\) before substituting back into each expression. - Check that the resulting angles have the required total.

1. Complementary angles sum to \(90^\circ\), so \((3x+6)+(x+12)=90\). 2. Combine like terms: \(4x+18=90\), so \(4x=72\) and \(x=18\). 3. The angles are \(3(18)+6=60^\circ\) and \(18+12=30^\circ\). 4. Check: \(60^\circ+30^\circ=90^\circ\).

\(x=18\); the angles are \(60^\circ\) and \(30^\circ\).

- Separate the relationships “vertical” and “adjacent.” - Which pair is congruent, and which pair forms a straight angle? - Use the relationship before doing any subtraction.

1. Vertical angles are congruent, so the vertical angle also measures \(68^\circ\). 2. Adjacent angles at the intersection form linear pairs with the \(68^\circ\) angle. 3. Each adjacent angle measures \(180^\circ-68^\circ=112^\circ\). 4. Priya is incorrect: the vertical angle is congruent to the given angle, while an adjacent linear-pair angle is supplementary to it.

Priya is incorrect. The vertical angle is \(68^\circ\), and each adjacent angle is \(112^\circ\).

- Write the target total for a complementary pair and for a supplementary pair. - “Congruent angles” means their measures are equal.

1. A complementary angle has measure \(90^\circ-42^\circ=48^\circ\). 2. A supplementary angle has measure \(180^\circ-42^\circ=138^\circ\). 3. A congruent angle has the same measure, \(42^\circ\).

Complementary: \(48^\circ\) Supplementary: \(138^\circ\) Congruent: \(42^\circ\)

- Which total belongs to complementary angles? - Which total belongs to supplementary angles? - Check which of Nora's calculations actually matches her result.

1. Nora used the total for supplementary angles, \(180^\circ\), instead of the total for complementary angles, \(90^\circ\). 2. The complement is \(90^\circ-63^\circ=27^\circ\). 3. The supplement is \(180^\circ-63^\circ=117^\circ\).

Nora used the supplementary total instead of the complementary total. The complement is \(27^\circ\), and the supplement is \(117^\circ\).

- Let the smaller angle be \(x\), and express the larger angle in terms of \(x\). - What is true about alternate interior angles when lines are parallel? - The two adjacent angles must continue to total \(180^\circ\).

1. Let \(x\) be the acute angle. The adjacent obtuse angle is \(2x\). 2. Since the angles form a linear pair, \(x + 2x = 180^\circ\). Thus \(3x = 180^\circ\), so \(x = 60^\circ\). The obtuse angle is \(120^\circ\). 3. Alternate interior angles formed by parallel lines are congruent. Therefore, the requested sum is \(60^\circ + 60^\circ = 120^\circ\). 4. The acute and obtuse angles remain a linear pair. If the acute angle decreases to \(50^\circ\), the obtuse angle becomes \(180^\circ - 50^\circ = 130^\circ\). It increases by \(10^\circ\).

a) The acute angle is \(60^\circ\), and the obtuse angle is \(120^\circ\). b) \(120^\circ\) c) The obtuse angle increases by \(10^\circ\), from \(120^\circ\) to \(130^\circ\).

- Plot the points and draw the first ray before adding the second rays. - Opposite rays form a straight angle. - Compare horizontal, vertical, and equal-change diagonal directions.

1. Ray \(\overrightarrow{SA}\) is vertical and points upward. 2. The opposite ray points vertically downward, so the two rays form a straight angle. Thus, \(\beta=180^\circ\). 3. From \(S\) to \(Q\), the change is \(4\) units left and \(4\) units up. This equal-change diagonal makes a \(45^\circ\) angle with the vertical ray, so \(\beta=45^\circ\). 4. Ray \(\overrightarrow{SP}\) points horizontally left. A horizontal ray and a vertical ray are perpendicular, so \(\beta=90^\circ\).

a) \(180^\circ\) b) \(45^\circ\) c) \(90^\circ\)

- Plot each named point and draw the rays before calculating an angle. - Equal horizontal and vertical changes indicate a \(45^\circ\) diagonal direction. - Perpendicular rays form a right angle, and opposite rays form a straight angle.

1. Ray \(\overrightarrow{SA}\) is horizontal and points to the right. 2. From \(S\) to \(P\), the change is \(2\) units left and \(2\) units up. This northwest diagonal has direction \(135^\circ\) from the positive horizontal direction, so \(\gamma=135^\circ\). 3. A downward ray perpendicular to \(\overrightarrow{SA}\) is vertical. Therefore, \(\gamma=90^\circ\). It follows the line \(x=2\) and meets the x-axis at \((2, 0)\). 4. Ray \(\overrightarrow{SQ}\) points horizontally left, exactly opposite \(\overrightarrow{SA}\). Therefore, \(\gamma=180^\circ\).

a) \(135^\circ\) b) \(90^\circ\); the ray meets the x-axis at \((2, 0)\). c) \(180^\circ\)

- What is the sum of two angles in a linear pair? - Represent the ratio \(4:5\) as \(4x\) and \(5x\). - How many equal ratio parts make up the full \(180^\circ\)?

1. Angles in a linear pair total \(180^\circ\). 2. Represent the angles as \(4x\) and \(5x\). Then \(4x + 5x = 180^\circ\). 3. Solve: \(9x = 180^\circ\), so \(x = 20^\circ\). 4. The angles are \(4 \cdot 20^\circ = 80^\circ\) and \(5 \cdot 20^\circ = 100^\circ\).

The angles measure \(80^\circ\) and \(100^\circ\).

- What is the sum of adjacent angles formed by two intersecting lines? - What relationship do vertical angles have? - Write one angle as \(36^\circ\) more than the other.

1. Adjacent angles formed by intersecting lines make a linear pair, so \(\alpha + \beta = 180^\circ\). 2. The given relationship is \(\alpha = \beta + 36^\circ\). 3. Substitute: \((\beta + 36^\circ) + \beta = 180^\circ\). 4. Solve: \(2\beta + 36^\circ = 180^\circ\), so \(2\beta = 144^\circ\) and \(\beta = 72^\circ\). 5. Then \(\alpha = 72^\circ + 36^\circ = 108^\circ\). 6. Vertical angles are congruent, so there are two \(72^\circ\) angles and two \(108^\circ\) angles.

There are two angles measuring \(72^\circ\) and two angles measuring \(108^\circ\).

- Describe the angle positions created by each transversal. - What is true about corresponding angles when two lines are parallel? - What is the sum of a linear pair?

1. The \(35^\circ\) angle and \(\gamma\) are corresponding angles formed by transversal \(s\). Since \(g \parallel h\), \(\gamma = 35^\circ\). 2. The \(120^\circ\) angle corresponds to a \(120^\circ\) angle at the other parallel line. That angle and \(\epsilon\) form a linear pair. 3. Therefore, \(\epsilon = 180^\circ - 120^\circ = 60^\circ\).

\(\gamma = 35^\circ\) and \(\epsilon = 60^\circ\).

- Use corresponding angle relationships on the parallel lines. - Identify the triangle formed by the three intersecting lines. - What is the sum of a triangle’s interior angles? - Use a linear pair to find the remaining angle.

1. The \(50^\circ\) angle and \(\gamma\) are corresponding angles. Since \(g \parallel h\), \(\gamma = 50^\circ\). 2. In the triangle formed by \(g\), \(s\), and \(t\), \(\alpha + \gamma + 75^\circ = 180^\circ\). 3. Substitute \(\gamma = 50^\circ\): \(\alpha + 50^\circ + 75^\circ = 180^\circ\), so \(\alpha = 55^\circ\). 4. At the other parallel line, the angle corresponding to \(\alpha\) also measures \(55^\circ\). It forms a linear pair with \(\beta\), so \(\beta = 180^\circ - 55^\circ = 125^\circ\).

\(\alpha = 55^\circ\), \(\beta = 125^\circ\), and \(\gamma = 50^\circ\).

- What relationship do vertical angles have? - What is the sum of a linear pair? - What is the sum of the interior angles of a triangle?

1. The \(65^\circ\) angle and \(\alpha\) are vertical angles, so \(\alpha = 65^\circ\). 2. The \(120^\circ\) angle and \(\beta\) form a linear pair, so \(\beta = 180^\circ - 120^\circ = 60^\circ\). 3. The three interior angles of the triangle total \(180^\circ\). Therefore, \(\gamma = 180^\circ - 65^\circ - 60^\circ = 55^\circ\).

\(\alpha = 65^\circ\), \(\beta = 60^\circ\), and \(\gamma = 55^\circ\).

- What does an angle bisector do? - Use corresponding or alternate interior angles formed by the parallel lines. - Which angle forms a linear pair with \(\beta\)?

1. Since \(w\) is an angle bisector, the two angles between \(g\) and \(t\) are congruent. The given half is \(28^\circ\), so \(\delta = 28^\circ\). 2. The full angle between \(g\) and \(t\) is \(28^\circ + 28^\circ = 56^\circ\). 3. The corresponding angle at line \(h\) also measures \(56^\circ\). It forms a linear pair with \(\beta\), so \(\beta = 180^\circ - 56^\circ = 124^\circ\). 4. The \(28^\circ\) angle formed by \(g\) and \(w\) is alternate interior to \(\gamma\). Therefore, \(\gamma = 28^\circ\).

\(\delta = 28^\circ\), \(\beta = 124^\circ\), and \(\gamma = 28^\circ\).

- Use corresponding and alternate interior angle relationships. - Find the interior angle at \(C\) by using a linear pair. - What is the sum of the interior angles of triangle \(ACE\)?

1. The \(60^\circ\) angle and \(\beta\) are alternate interior angles formed by transversal \(s\). Therefore, \(\beta = 60^\circ\). 2. At \(C\), the angle corresponding to the given \(130^\circ\) angle measures \(130^\circ\). The interior angle of triangle \(ACE\) at \(C\) is its linear-pair angle: \(180^\circ - 130^\circ = 50^\circ\). 3. In triangle \(ACE\), \(\alpha = 180^\circ - 60^\circ - 50^\circ = 70^\circ\).

\(\alpha = 70^\circ\) and \(\beta = 60^\circ\).

- What is the measure of a straight angle? - First find the full angle that is bisected. - What does an angle bisector do to an angle?

1. The given \(60^\circ\) angle and the angle between \(h\) and the left-hand ray of \(g\) form a linear pair. 2. That angle measures \(180^\circ - 60^\circ = 120^\circ\). 3. Since \(k\) bisects the \(120^\circ\) angle, \(\alpha = \beta = 120^\circ \div 2 = 60^\circ\).

\(\alpha = 60^\circ\) and \(\beta = 60^\circ\).

- Which corresponding or alternate angles can you identify? - What total do adjacent angles on a straight line have? - You may also identify the triangle formed by the two transversals and one parallel line.

1. By the corresponding and alternate angle relationships for parallel lines, the two angles adjacent to \(\alpha\) along line \(h\) measure \(60^\circ\) and \(50^\circ\). 2. These three angles form a straight angle, so \(60^\circ + \alpha + 50^\circ = 180^\circ\). 3. Therefore, \(\alpha = 180^\circ - 60^\circ - 50^\circ = 70^\circ\).

\(\alpha = 70^\circ\)

- What total do the three angles below line \(g\) form? - Use alternate interior angle relationships between \(g\) and \(h\). - Which known angle forms a linear pair with \(\gamma\)?

1. The three angles below line \(g\) form a straight angle. Therefore, \(\alpha = 180^\circ - 70^\circ - 50^\circ = 60^\circ\). 2. The \(70^\circ\) angle and \(\beta\) are alternate interior angles, so \(\beta = 70^\circ\). 3. The angle alternate interior to the given \(50^\circ\) angle measures \(50^\circ\). It forms a linear pair with \(\gamma\), so \(\gamma = 180^\circ - 50^\circ = 130^\circ\).

\(\alpha = 60^\circ\), \(\beta = 70^\circ\), and \(\gamma = 130^\circ\).

- What relationship connects an interior angle and its adjacent exterior angle? - What does the exterior angle theorem state? - Test the equations with a triangle whose angles have three different measures.

1. By the exterior angle theorem, an exterior angle of a triangle equals the sum of the two remote interior angles. Therefore, \(\delta = \alpha + \beta\), so statement a) is true. 2. Angles \(\gamma\) and \(\delta\) form a linear pair, so \(\delta + \gamma = 180^\circ\). Equivalently, \(\delta = 180^\circ - \gamma\). Therefore, statements b) and c) are true. 3. The equation \(\delta = \beta + \gamma\) would require \(\alpha = \gamma\), which is not true for every triangle. Therefore, statement d) is not always true.

d) \(\delta = \beta + \gamma\) is not always true.

- What relationship do vertical angles have? - What total do adjacent angles along a straight line form? - Find the vertical angles first.

1. The \(40^\circ\) angle and \(\alpha\) are vertical angles, so \(\alpha = 40^\circ\). 2. The \(75^\circ\) angle and \(\beta\) are vertical angles, so \(\beta = 75^\circ\). 3. The angles \(\alpha\), \(\beta\), and \(\gamma\) form a straight angle. Therefore, \(\gamma = 180^\circ - 40^\circ - 75^\circ = 65^\circ\).

\(\alpha = 40^\circ\), \(\beta = 75^\circ\), and \(\gamma = 65^\circ\).

- What is the sum of the interior angles of a triangle? - After finding the interior angle at \(C\), how can you use its linear pair? - Which theorem relates an exterior angle directly to the two remote interior angles?

1. Method 1: The triangle angle sum gives the interior angle at \(C\): \(180^\circ - 55^\circ - 65^\circ = 60^\circ\). This angle and \(\alpha\) form a linear pair, so \(m\angle \alpha = 180^\circ - 60^\circ = 120^\circ\). 2. Method 2: By the exterior angle theorem, an exterior angle equals the sum of the two remote interior angles. Therefore, \(m\angle \alpha = 55^\circ + 65^\circ = 120^\circ\).

\(m\angle \alpha = 120^\circ\)

- Identify alternate interior and corresponding angles. - Which unknown angle is congruent to the given \(55^\circ\) angle? - Which unknown angle forms a linear pair with a \(115^\circ\) angle?

1. The angle \(\gamma\) is alternate interior to the given \(55^\circ\) angle. Since \(g \parallel h\), \(\gamma = 55^\circ\). 2. The angle corresponding to the given \(115^\circ\) angle forms a linear pair with \(\delta\). 3. Therefore, \(\delta = 180^\circ - 115^\circ = 65^\circ\).

\(\gamma = 55^\circ\) and \(\delta = 65^\circ\).

- Recall what must be true about consecutive angles in every parallelogram. - Recall what must be true about opposite angles in every parallelogram. - For the exterior-angle case, first determine the adjacent interior angle.

1. In a parallelogram, consecutive interior angles are supplementary. For a), \(68^\circ+110^\circ=178^\circ\), so these measurements cannot occur in a parallelogram. 2. In a parallelogram, opposite angles are congruent. For b), opposite angles of \(80^\circ\) are consistent with a parallelogram; the other pair could each be \(100^\circ\). Thus at least one parallelogram can have these measurements. 3. For c), the interior angle adjacent to the \(105^\circ\) exterior angle is \(180^\circ-105^\circ=75^\circ\). 4. Its opposite interior angle is given as \(73^\circ\). Since \(75^\circ\ne73^\circ\), these measurements cannot occur in a parallelogram.

a) No; \(68^\circ+110^\circ\ne180^\circ\). b) Yes; a parallelogram can have opposite angles of \(80^\circ\) and the other opposite pair of \(100^\circ\). c) No; the interior angle next to the \(105^\circ\) exterior angle is \(75^\circ\), not equal to its opposite \(73^\circ\) angle.

- What relationship do vertical angles have? - What is the sum of a linear pair? - How are corresponding angles related when lines are parallel?

1. The given \(55^\circ\) angle and \(\alpha\) are vertical angles, so \(\alpha = 55^\circ\). 2. The angle corresponding to \(\alpha\) at the upper intersection also measures \(55^\circ\). It forms a linear pair with \(\beta\), so \(\beta = 180^\circ - 55^\circ = 125^\circ\). 3. The angles \(\beta\) and \(\gamma\) are vertical angles, so \(\gamma = 125^\circ\).

\(\alpha = 55^\circ\), \(\beta = 125^\circ\), and \(\gamma = 125^\circ\).

- Describe the lines and intersections shown in the diagram. - Which adjacent angles form a straight angle? - What angle relationships occur when a transversal crosses parallel lines? - Begin with the angle you can find directly, and then work one intersection at a time.

1. At the intersection on line \(g\), the three adjacent angles form a straight angle. Therefore, \(60^\circ + \alpha + 45^\circ = 180^\circ\), so \(\alpha = 75^\circ\). 2. The angle adjacent to \(\beta\) is corresponding to the given \(60^\circ\) angle, so it also measures \(60^\circ\). These two angles form a linear pair, so \(\beta = 180^\circ - 60^\circ = 120^\circ\). 3. The angle adjacent to \(\gamma\) is corresponding to the given \(45^\circ\) angle, so it also measures \(45^\circ\). These two angles form a linear pair, so \(\gamma = 180^\circ - 45^\circ = 135^\circ\).

\(\alpha = 75^\circ\), \(\beta = 120^\circ\), and \(\gamma = 135^\circ\)

- First transfer the given angle from \(h\) to the parallel line \(g\). - Then focus on the triangle below \(g\). - What is the sum of the interior angles of a triangle?

1. Because \(g \parallel h\), the interior angle at the left intersection with \(g\) has the same measure as the given \(50^\circ\) angle. 2. The triangle below \(g\) therefore has two known interior angles: \(50^\circ\) and \(80^\circ\). 3. Using the triangle angle sum, \(\beta = 180^\circ - 50^\circ - 80^\circ = 50^\circ\).

\(\beta = 50^\circ\)

- What angle relationships occur when a transversal crosses two parallel lines? - What is true about adjacent angles that form a straight line? - Review corresponding, alternate interior, and vertical angles. - Can you first find the angle next to the given angle? - How are the angles at the upper intersection related to those at the lower intersection?

1. The angle supplementary to the given \(115^\circ\) angle measures \(180^\circ - 115^\circ = 65^\circ\). 2. Angle \(\alpha\) corresponds to this \(65^\circ\) angle, so \(\alpha = 65^\circ\). 3. Angles \(\alpha\) and \(\beta\) are vertical angles, so \(\beta = \alpha = 65^\circ\).

\(\alpha = 65^\circ\) and \(\beta = 65^\circ\)

- What angle relationships occur when a transversal crosses two parallel lines? - Review corresponding and alternate interior angles. - What is the sum of the interior angles of a triangle? - Which shown angle is supplementary to the interior angle at \(B\)?

1. Because \(g \parallel h\), \(\alpha\) is alternate interior to the given \(50^\circ\) angle. Therefore, \(\alpha = 50^\circ\). 2. The \(130^\circ\) angle at \(D\) corresponds to the exterior angle at \(B\). The interior angle of triangle \(ABC\) at \(B\) is supplementary to that angle, so it measures \(180^\circ - 130^\circ = 50^\circ\). 3. The interior angles of triangle \(ABC\) total \(180^\circ\). Therefore, \(\gamma = 180^\circ - 50^\circ - 50^\circ = 80^\circ\).

\(\alpha = 50^\circ\) and \(\gamma = 80^\circ\)

- What angle relationships occur when other lines cross two parallel lines? - Review alternate interior and corresponding angles. - How could an auxiliary line through \(P\), parallel to \(g\) and \(h\), help? - Can you identify a triangle whose angle sum will help?

1. Because \(g \parallel h\), \(\beta\) is alternate interior to the given \(40^\circ\) angle at \(A\). Therefore, \(\beta = 40^\circ\). 2. The angle at \(B\) that is exterior to triangle \(ABP\) corresponds to the given \(125^\circ\) angle at \(D\). The interior angle at \(B\) is its supplement, so it measures \(180^\circ - 125^\circ = 55^\circ\). 3. In triangle \(ABP\), \(\alpha = 180^\circ - 40^\circ - 55^\circ = 85^\circ\).

\(\alpha = 85^\circ\) and \(\beta = 40^\circ\)

- Angles that form a straight angle have a sum of \(180^\circ\). - What angle relationships occur when a transversal crosses parallel lines? Think about corresponding and alternate interior angles. - Can you identify a triangle in the figure and use its angle sum? - What is true about vertical angles?

1. Angles \(\alpha\), \(55^\circ\), and \(60^\circ\) are adjacent along line \(g\) and form a straight angle. Therefore, \(\alpha = 180^\circ - 55^\circ - 60^\circ = 65^\circ\). 2. Because \(g \parallel h\), \(\delta\) is alternate interior to \(\alpha\). Therefore, \(\delta = 65^\circ\). 3. Angle \(\varepsilon\) corresponds to the given \(60^\circ\) angle. Therefore, \(\varepsilon = 60^\circ\).

\(\alpha = 65^\circ\), \(\delta = 65^\circ\), and \(\varepsilon = 60^\circ\)

- What angle relationships occur when a transversal crosses parallel lines? - Which corresponding or alternate interior angles can you identify? - What is the sum of the interior angles of a triangle? - Use the large triangle to find one unknown angle. - Which unknown angle forms a linear pair with a known angle?

1. Because \(g \parallel h\), \(\beta\) corresponds to the given \(50^\circ\) angle. Therefore, \(\beta = 50^\circ\). 2. In the large triangle, \(\alpha = 180^\circ - 50^\circ - 60^\circ = 70^\circ\). 3. The interior angle at the lower intersection on the right is corresponding to the given \(60^\circ\) angle, so it also measures \(60^\circ\). 4. This \(60^\circ\) angle and \(\gamma\) form a linear pair. Therefore, \(\gamma = 180^\circ - 60^\circ = 120^\circ\).

\(\alpha = 70^\circ\), \(\beta = 50^\circ\), and \(\gamma = 120^\circ\)

- Which angles are directly opposite each other? What is true about those angles? - Which adjacent angles form a straight angle, and what is their sum? - First find the angles determined directly by vertical angle relationships. - Then use a straight angle to find the remaining measure.

1. Angle \(\gamma\) is vertical to the given \(55^\circ\) angle, so \(\gamma = 55^\circ\). 2. Angle \(\alpha\) is vertical to the given \(70^\circ\) angle, so \(\alpha = 70^\circ\). 3. The angles \(55^\circ\), \(\alpha\), and \(\beta\) form a straight angle. Therefore, \(\beta = 180^\circ - 55^\circ - 70^\circ = 55^\circ\). 4. Angle \(\delta\) is vertical to \(\beta\), so \(\delta = 55^\circ\).

\(\alpha = 70^\circ\), \(\beta = 55^\circ\), \(\gamma = 55^\circ\), and \(\delta = 55^\circ\)

- Start with the large triangle \(ABC\). What is the sum of its interior angles? - Because \(DE \parallel BC\), which corresponding or alternate interior angles can you identify? - How are \(\beta\) and \(\angle ADE\) related along line \(AB\)? - What is the sum of a linear pair?

1. In triangle \(ABC\), \(\alpha = 180^\circ - 50^\circ - 60^\circ = 70^\circ\). 2. Because \(DE \parallel BC\), \(\angle ADE\) corresponds to \(\angle ABC\). Therefore, \(\angle ADE = 50^\circ\). 3. Angles \(\beta\) and \(\angle ADE\) form a linear pair along line \(AB\). Thus, \(\beta = 180^\circ - 50^\circ = 130^\circ\).

\(\alpha = 70^\circ\) and \(\beta = 130^\circ\)

- Which lines are parallel? - What angle relationships occur when a transversal crosses parallel lines? - Can you identify a triangle in the diagram and use its angle sum? - Which angles form linear pairs? - Begin with the angles directly related to the given measures.

1. The angle corresponding to the given \(70^\circ\) angle is supplementary to \(\beta\). Therefore, \(\beta = 180^\circ - 70^\circ = 110^\circ\). 2. The two transversals and line \(g\) form a triangle with base angles \(70^\circ\) and \(60^\circ\). Thus, \(\alpha = 180^\circ - 70^\circ - 60^\circ = 50^\circ\). 3. The angle corresponding to the given \(60^\circ\) angle is supplementary to \(\gamma\). Therefore, \(\gamma = 180^\circ - 60^\circ = 120^\circ\).

\(\alpha = 50^\circ\), \(\beta = 110^\circ\), and \(\gamma = 120^\circ\)

- What angle relationships occur when a transversal crosses parallel lines? - Think about alternate interior and corresponding angles. - What is true about vertical angles? - Use the triangle angle sum in one of the triangles.

1. Because \(g \parallel h\), \(\gamma\) is alternate interior to the given \(45^\circ\) angle. Therefore, \(\gamma = 45^\circ\). 2. In triangle \(SBD\), the angles at \(B\) and \(D\) measure \(30^\circ\) and \(45^\circ\). Thus, \(\angle BSD = 180^\circ - 30^\circ - 45^\circ = 105^\circ\). 3. Angle \(\beta\) is vertical to \(\angle BSD\), so \(\beta = 105^\circ\).

\(\beta = 105^\circ\) and \(\gamma = 45^\circ\)

- What is the sum of the angles in a linear pair? - Express both \(\alpha\) and \(\alpha'\) in terms of \(\beta\). - What follows if two angles have the same measure?

1. Since \(\alpha\) and \(\beta\) form a linear pair, \(\alpha + \beta = 180^\circ\), so \(\alpha = 180^\circ - \beta\). 2. Since \(\alpha'\) and \(\beta\) also form a linear pair, \(\alpha' + \beta = 180^\circ\), so \(\alpha' = 180^\circ - \beta\). 3. Both angles equal \(180^\circ - \beta\), so \(\alpha = \alpha'\). This proves that vertical angles are congruent.

Both \(\alpha\) and \(\alpha'\) equal \(180^\circ - \beta\), so \(\alpha = \alpha'\).

- Focus first on transversal \(t\), which crosses the parallel lines at \(R\) and \(D\). - What is the relationship between the \(135^\circ\) angle at \(D\) and \(\alpha\)? - After finding \(\alpha\), use the triangle angle sum in triangle \(PQR\).

1. Along transversal \(t\), the \(135^\circ\) angle at \(D\) and \(\alpha\) are same-side interior angles between parallel lines \(a\) and \(b\). 2. Same-side interior angles are supplementary, so \(\alpha=180^\circ-135^\circ=45^\circ\). 3. In triangle \(PQR\), the interior angles are \(40^\circ\), \(45^\circ\), and \(\beta\). Thus, \(\beta=180^\circ-40^\circ-45^\circ=95^\circ\).

\(\alpha=45^\circ\) and \(\beta=95^\circ\)

- Use the triangle angle sum. - How are an interior angle and its adjacent exterior angle related? - Look for corresponding angles formed by the parallel lines.

1. The two transversals and line \(g\) form a triangle. Its two base angles are \(65^\circ\) and \(40^\circ\), so its interior angle at their intersection is \(180^\circ-65^\circ-40^\circ=75^\circ\). 2. The marked angle \(\beta\) is vertical to that \(75^\circ\) triangle angle, so \(\beta=75^\circ\). 3. Angles \(\alpha\) and \(\beta\) form a linear pair, so \(\alpha=180^\circ-75^\circ=105^\circ\). 4. At the lower intersection with transversal \(t\), the angle adjacent to the given \(40^\circ\) angle measures \(140^\circ\). That angle corresponds to \(\gamma\) because \(g\parallel h\). Therefore, \(\gamma=140^\circ\).

\(\alpha=105^\circ\), \(\beta=75^\circ\), and \(\gamma=140^\circ\)

- A linear pair has a sum of \(180^\circ\). - Use corresponding angles formed by the parallel lines.

1. Angle \(\alpha\) and the given \(115^\circ\) angle form a linear pair, so \(\alpha = 180^\circ - 115^\circ = 65^\circ\). 2. Because \(p \parallel q\), \(\gamma\) corresponds to the given \(48^\circ\) angle, so \(\gamma = 48^\circ\). 3. The two transversals and line \(p\) form a triangle. Therefore, \(\beta = 180^\circ - 65^\circ - 48^\circ = 67^\circ\).

\(\alpha = 65^\circ\), \(\beta = 67^\circ\), and \(\gamma = 48^\circ\)

- What do matching arc marks tell you? - First use the given exterior angles to find adjacent interior angles. - Check that the four interior angles total \(360^\circ\).

1. Angles \(\alpha\) and \(\beta\) each form a linear pair with a \(70^\circ\) exterior angle. Therefore, \(\alpha = \beta = 180^\circ - 70^\circ = 110^\circ\). 2. Angle \(\gamma\) forms a linear pair with the \(110^\circ\) exterior angle, so \(\gamma = 70^\circ\). 3. The matching arc marks show that \(\delta = \gamma = 70^\circ\). 4. The check is \(110^\circ + 110^\circ + 70^\circ + 70^\circ = 360^\circ\).

\(\alpha = 110^\circ\), \(\beta = 110^\circ\), \(\gamma = 70^\circ\), and \(\delta = 70^\circ\)

- What is the sum of the measures in a linear pair? - Express both unknown angles in terms of the same value \(x\).

1. Each unknown angle forms a linear pair with one of the given congruent angles. 2. Let each given angle measure \(x^\circ\). Then each unknown angle measures \(180^\circ - x^\circ\). 3. Since both unknown angles have the same measure, they are congruent.

Both unknown angles are supplementary to congruent angles, so each measures \(180^\circ - x^\circ\). Therefore, they are congruent.

- Which triangle sides act as transversals of the two parallel lines? - Which triangle angles match the two given angles by alternate interior angle relationships? - After finding two triangle angles, what total should all three interior angles have?

1. Because \(g \parallel h\), alternate interior angles are congruent. Therefore, \(\angle A = 50^\circ\) and \(\angle B = 60^\circ\). 2. Use the triangle angle sum: \(\angle C = 180^\circ - 50^\circ - 60^\circ = 70^\circ\).

\(\angle A = 50^\circ\), \(\angle B = 60^\circ\), and \(\angle C = 70^\circ\)

- The angles in a linear pair sum to \(180^\circ\). - Use the stated relationship between \(\alpha\) and \(\beta\) to write the supplementary-angle equation with one unknown.

1. A linear pair is supplementary, so \(\alpha + \beta = 180^\circ\). 2. Substitute \(\alpha = 90^\circ + \beta\): \((90^\circ + \beta) + \beta = 180^\circ\). 3. Solve: \(2\beta = 90^\circ\), so \(\beta = 45^\circ\). 4. Then \(\alpha = 90^\circ + 45^\circ = 135^\circ\).

\(\alpha = 135^\circ\) and \(\beta = 45^\circ\)

- Represent the smaller angle as one part and the larger angle as four equal parts of a \(180^\circ\) total. - Let one part be an unknown angle measure, then express the larger angle using that same unknown.

1. Let \(\beta\) be the smaller angle. Then \(\alpha = 4\beta\). 2. Since the angles form a linear pair, \(4\beta + \beta = 180^\circ\). 3. Solve: \(5\beta = 180^\circ\), so \(\beta = 36^\circ\). 4. Then \(\alpha = 4 \cdot 36^\circ = 144^\circ\).

The angle measures are \(36^\circ\) and \(144^\circ\).

- Into how many equal ratio parts is the \(180^\circ\) straight angle divided? - After finding the size of one ratio part, match one part to \(\alpha\) and five parts to \(\beta\).

1. The ratio \(1:5\) divides the \(180^\circ\) total into \(1 + 5 = 6\) equal parts. 2. One part measures \(180^\circ \div 6 = 30^\circ\). 3. Therefore, \(\alpha = 30^\circ\) and \(\beta = 5 \cdot 30^\circ = 150^\circ\).

\(\alpha = 30^\circ\) and \(\beta = 150^\circ\)

- What is the total measure of all angles around a point? - Once you know one angle, which angle is congruent to it? - What relationship gives the two adjacent angle measures?

1. All four angles around the intersection total \(360^\circ\), so the omitted angle measures \(360^\circ - 305^\circ = 55^\circ\). 2. Its vertical angle also measures \(55^\circ\). 3. Each adjacent angle forms a linear pair with a \(55^\circ\) angle, so each measures \(180^\circ - 55^\circ = 125^\circ\).

Two angles measure \(55^\circ\), and two angles measure \(125^\circ\).

- The four angles around the intersection total \(360^\circ\). - Write an equation comparing the unknown angle with the sum of the other three.

1. Let \(\alpha\) be the unknown angle. The other three angles have a total measure of \(360^\circ - \alpha\). 2. Write the equation \(\alpha = \frac{1}{4}(360^\circ - \alpha)\). 3. Solve: \(4\alpha = 360^\circ - \alpha\), so \(5\alpha = 360^\circ\) and \(\alpha = 72^\circ\).

\(\alpha = 72^\circ\)

- Let the shared middle angle between \(b\) and \(c\) have measure \(x^\circ\). - Express each of the two target angles as the remainder of a right angle. - What follows if both expressions are the same?

1. Let the angle between rays \(b\) and \(c\) have measure \(x^\circ\). 2. Since the angle from \(a\) to \(c\) is a right angle, the angle between \(a\) and \(b\) measures \(90^\circ - x^\circ\). 3. Since the angle from \(b\) to \(d\) is also a right angle, the angle between \(c\) and \(d\) measures \(90^\circ - x^\circ\). 4. The two angles have the same measure, so they are congruent.

Both angles are what remains after subtracting the shared angle between \(b\) and \(c\) from \(90^\circ\), so they are congruent.

- What angle measure is required for two rays to form a straight line? - What does each bisector do to a \(60^\circ\) angle? - Include the angle supplementary to \(60^\circ\) between the two bisected angles.

1. The opposite vertical angle also measures \(60^\circ\). 2. Each angle bisector divides a \(60^\circ\) angle into two \(30^\circ\) angles. 3. An angle adjacent to a \(60^\circ\) angle measures \(180^\circ - 60^\circ = 120^\circ\). 4. The angle from \(w_1\) to \(w_2\) measures \(30^\circ + 120^\circ + 30^\circ = 180^\circ\). 5. Therefore, \(w_1\) and \(w_2\) are opposite rays and lie on the same line.

The angle between \(w_1\) and \(w_2\) is \(30^\circ + 120^\circ + 30^\circ = 180^\circ\), so the rays lie on one straight line.

- Think about multiples of \(35^\circ\). - A straight angle measures \(180^\circ\). - For the smallest target, find a nearby multiple of \(35^\circ\) and use its supplement.

1. a) Three \(35^\circ\) angles total \(3 \cdot 35^\circ = 105^\circ\). 2. b) Four \(35^\circ\) angles total \(4 \cdot 35^\circ = 140^\circ\). 3. c) Five \(35^\circ\) angles total \(5 \cdot 35^\circ = 175^\circ\). The supplement in a straight angle is \(180^\circ - 175^\circ = 5^\circ\).

a) Use three \(35^\circ\) angles. b) Use four \(35^\circ\) angles. c) Use the supplement of five \(35^\circ\) angles: \(180^\circ - 175^\circ = 5^\circ\).

- What is the sum of adjacent angles formed by two intersecting lines? - Express the smaller angle in terms of the larger angle. - After finding one adjacent pair, which opposite angles must match them?

1. Let \(\alpha\) be the smaller angle and \(\beta\) the adjacent larger angle. Then \(\alpha + \beta = 180^\circ\) and \(\alpha = \beta - 60^\circ\). 2. Substitute: \((\beta - 60^\circ) + \beta = 180^\circ\). 3. Solve: \(2\beta = 240^\circ\), so \(\beta = 120^\circ\) and \(\alpha = 60^\circ\). 4. Vertical angles are congruent, so the four angle measures alternate between \(60^\circ\) and \(120^\circ\).

The four angles measure \(60^\circ\), \(120^\circ\), \(60^\circ\), and \(120^\circ\).

- What is the total angle measure around a point? - Replace \(\gamma\) and \(\delta\) using the given equalities. - What geometric conclusion follows from a \(180^\circ\) angle?

1. Angles around a point sum to \(360^\circ\), so \(\alpha + \beta + \gamma + \delta = 360^\circ\). 2. Substituting \(\gamma = \alpha\) and \(\delta = \beta\) gives \(\alpha + \beta + \alpha + \beta = 360^\circ\). 3. Therefore, \(2(\alpha + \beta) = 360^\circ\), so \(\alpha + \beta = 180^\circ\). 4. The nonshared sides of adjacent angles \(\alpha\) and \(\beta\) are therefore opposite rays, so rays \(a\) and \(c\) lie on one line. Also, \(\beta + \gamma = \beta + \alpha = 180^\circ\), so rays \(b\) and \(d\) are opposite rays and lie on another line. 5. Therefore, the four rays form two intersecting lines.

The angle sum gives \(2(\alpha + \beta) = 360^\circ\), so \(\alpha + \beta = 180^\circ\). Thus, rays \(a\) and \(c\) are opposite rays. Since \(\beta + \gamma = \beta + \alpha = 180^\circ\), rays \(b\) and \(d\) are also opposite rays. Therefore, the four rays form two intersecting lines.

- Which opposite sides of a parallelogram are parallel? - Use diagonal \(AC\) as a transversal to identify alternate interior angles. - Then use the triangle angle sum in triangle \(ABC\).

1. Since opposite sides of a parallelogram are parallel, alternate interior angles give \(\angle ACB = \angle CAD = 40^\circ\) and \(\angle ACD = \angle BAC = 25^\circ\). 2. In triangle \(ABC\), \(\beta = 180^\circ - 25^\circ - 40^\circ = 115^\circ\).

a) \(\angle ACB = 40^\circ\) and \(\angle ACD = 25^\circ\) b) \(\beta = 115^\circ\)

- Use the complementary relationship first. - Then use the supplementary relationship that shares \(\beta\). - Compare the first and last angle only after both are known.

1. Since \(\alpha\) and \(\beta\) are complementary, \(\beta=90^\circ-34^\circ=56^\circ\). 2. Since \(\beta\) and \(\gamma\) are supplementary, \(\gamma=180^\circ-56^\circ=124^\circ\). 3. The difference is \(124^\circ-34^\circ=90^\circ\), so \(\gamma\) is \(90^\circ\) greater than \(\alpha\).

\(\beta=56^\circ\), \(\gamma=124^\circ\), and \(\gamma=\alpha+90^\circ\).

- Which adjacent angles form a straight angle? - What is the sum of the interior angles of a triangle? - Which opposite angles are congruent when two lines intersect? - Find the missing angle in the left triangle before working with the right triangle.

1. The \(75^\circ\) angle and \(\alpha\) form a linear pair, so \(\alpha = 180^\circ - 75^\circ = 105^\circ\). 2. The interior angles of triangle \(ABS\) total \(180^\circ\). Therefore, \(\beta = 180^\circ - 75^\circ - 65^\circ = 40^\circ\). 3. The angles \(\angle ASB\) and \(\angle CSD\) are vertical angles, so \(\angle CSD = \beta = 40^\circ\). 4. The interior angles of triangle \(CDS\) total \(180^\circ\). Therefore, \(\gamma = 180^\circ - 40^\circ - 50^\circ = 90^\circ\).

\(\alpha = 105^\circ\), \(\beta = 40^\circ\), and \(\gamma = 90^\circ\)

- Consider a line through the vertex of \(\alpha\) parallel to \(g\) and \(h\). - How does that line split \(\alpha\) into two smaller angles? - Which alternate interior angle relationships can you use?

1. Consider an auxiliary line through the vertex of \(\alpha\) parallel to \(g\) and \(h\). 2. This line divides \(\alpha\) into two angles, so \(\alpha = \alpha_1 + \alpha_2\). 3. By alternate interior angle relationships, \(\alpha_1 = 35^\circ\) and \(\alpha_2 = 40^\circ\). 4. Therefore, \(\alpha = 35^\circ + 40^\circ = 75^\circ\).

\(\alpha = 75^\circ\)

- What is true about vertical angles and about adjacent angles that form a straight line? - Consider each triangle separately. What is the sum of its interior angles? - How can the angles at \(S\) connect the two triangles? - Look for angles that form a linear pair.

1. The given \(110^\circ\) angle and the interior angle of the left triangle at \(S\) form a linear pair. That interior angle is \(180^\circ - 110^\circ = 70^\circ\). 2. In the left triangle, \(\alpha = 180^\circ - 45^\circ - 70^\circ = 65^\circ\). 3. The interior angles at \(S\) in the two triangles are vertical angles, so the angle at \(S\) in the right triangle is also \(70^\circ\). 4. In the right triangle, \(\beta = 180^\circ - 70^\circ - 45^\circ = 65^\circ\). 5. The \(45^\circ\) angle and \(\gamma\) form a linear pair, so \(\gamma = 180^\circ - 45^\circ = 135^\circ\).

\(\alpha = 65^\circ\), \(\beta = 65^\circ\), and \(\gamma = 135^\circ\)

- Which angles are opposite each other when two lines intersect? - Which groups of adjacent angles form a straight angle? - Start with the two given measures and use vertical angles to find their opposites. - Check that all six angles sum to \(360^\circ\).

1. Claim 1 is true because \(\alpha\), \(\beta\), and \(\gamma\) form a straight angle. 2. Claim 2 is false. Angle \(\delta\) is vertical to \(\alpha\), so \(\delta = 60^\circ\), while \(\beta = 65^\circ\). 3. Claim 3 is true because \(\gamma\) and \(\varphi\) are vertical angles. 4. Claim 4 is true. Vertical angles give \(\delta = \alpha = 60^\circ\) and \(\varepsilon = \beta = 65^\circ\), so \(60^\circ + 65^\circ + 60^\circ + 65^\circ = 250^\circ\). 5. Since \(\alpha + \beta + \gamma = 180^\circ\), \(\gamma = 180^\circ - 60^\circ - 65^\circ = 55^\circ\). The vertical angle pairs then give \(\delta = 60^\circ\), \(\varepsilon = 65^\circ\), and \(\varphi = 55^\circ\).

a) Claims 1, 3, and 4 are true. Claim 2 is false. b) \(\alpha = 60^\circ\), \(\beta = 65^\circ\), \(\gamma = 55^\circ\), \(\delta = 60^\circ\), \(\varepsilon = 65^\circ\), and \(\varphi = 55^\circ\).

- Which triangles can you identify in the diagram? - What is the sum of the interior angles of a triangle? - What angle relationships occur where two lines intersect? - What is the sum of adjacent angles that form a straight line? - Which unknown can you find first using one triangle?

1. In the large triangle, \(\alpha = 180^\circ - 60^\circ - 55^\circ = 65^\circ\). 2. In the upper small triangle, the angle adjacent to the given \(135^\circ\) angle is \(180^\circ - 135^\circ = 45^\circ\). 3. Therefore, \(\beta = 180^\circ - 55^\circ - 45^\circ = 80^\circ\). 4. Angles \(\beta\) and \(\gamma\) form a linear pair, so \(\gamma = 180^\circ - 80^\circ = 100^\circ\).

\(\alpha = 65^\circ\), \(\beta = 80^\circ\), and \(\gamma = 100^\circ\)

- Consider an auxiliary line through vertex \(V\) parallel to \(g\) and \(h\). - Which corresponding, alternate interior, or supplementary angle relationships can you use? - How does the auxiliary line divide the unknown angle into two smaller angles?

1. Consider an auxiliary line through vertex \(V\) parallel to \(g\) and \(h\). This divides \(\gamma\) into two smaller angles. 2. The lower part is alternate interior to the given \(50^\circ\) angle, so it measures \(50^\circ\). 3. The supplement of the given \(120^\circ\) angle is \(60^\circ\). The upper part is alternate interior to this \(60^\circ\) angle. 4. Therefore, \(\gamma = 50^\circ + 60^\circ = 110^\circ\).

\(\gamma = 110^\circ\)